Transition to turbulence and Singularity in Boundary Layer Theory

We compute the solutions of Prandtl’s and Navier- Stokes equations for the two dimensional flow induced by an array of periodic rectilinear vortices interacting with a boundary in the halfplane. This initial datum develops, in a finite time, a separation singularity for Prandtl’s equation. We investigate the different stages of unsteady separation in Navier-Stokes solutions for various Reynolds numbers. We show the presence of a large- scale interaction between viscous boundary layer and inviscid outer flow in all Re regimes, while the presence of a small-scale interaction is visible only for moderate-high Re numbers. We also investigate the asymptotic validity of boundary layer theory in the limit of infinite Re numbers. The numerical solutions are computed using an efficient parallel spectral-finite differences scheme for both Prandtl and Navier-Stokes, with a focusing of the grid points close to the boundary.